| L(s) = 1 | + (2.02 + 0.562i)2-s + (2.05 + 1.24i)4-s + (3.69 − 0.143i)5-s + (0.392 + 0.0613i)7-s + (0.582 + 0.617i)8-s + (7.55 + 1.78i)10-s + (−0.272 − 1.99i)11-s + (−0.138 + 0.202i)13-s + (0.758 + 0.344i)14-s + (−1.41 − 2.67i)16-s + (−3.38 + 1.69i)17-s + (0.121 + 2.08i)19-s + (7.78 + 4.29i)20-s + (0.572 − 4.18i)22-s + (−0.303 + 0.785i)23-s + ⋯ |
| L(s) = 1 | + (1.42 + 0.397i)2-s + (1.02 + 0.621i)4-s + (1.65 − 0.0641i)5-s + (0.148 + 0.0231i)7-s + (0.206 + 0.218i)8-s + (2.38 + 0.565i)10-s + (−0.0821 − 0.601i)11-s + (−0.0385 + 0.0561i)13-s + (0.202 + 0.0921i)14-s + (−0.352 − 0.669i)16-s + (−0.820 + 0.411i)17-s + (0.0278 + 0.479i)19-s + (1.74 + 0.960i)20-s + (0.121 − 0.892i)22-s + (−0.0632 + 0.163i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 - 0.375i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.926 - 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.88301 + 0.756902i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.88301 + 0.756902i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (-2.02 - 0.562i)T + (1.71 + 1.03i)T^{2} \) |
| 5 | \( 1 + (-3.69 + 0.143i)T + (4.98 - 0.387i)T^{2} \) |
| 7 | \( 1 + (-0.392 - 0.0613i)T + (6.66 + 2.13i)T^{2} \) |
| 11 | \( 1 + (0.272 + 1.99i)T + (-10.5 + 2.94i)T^{2} \) |
| 13 | \( 1 + (0.138 - 0.202i)T + (-4.68 - 12.1i)T^{2} \) |
| 17 | \( 1 + (3.38 - 1.69i)T + (10.1 - 13.6i)T^{2} \) |
| 19 | \( 1 + (-0.121 - 2.08i)T + (-18.8 + 2.20i)T^{2} \) |
| 23 | \( 1 + (0.303 - 0.785i)T + (-17.0 - 15.4i)T^{2} \) |
| 29 | \( 1 + (6.13 + 4.38i)T + (9.38 + 27.4i)T^{2} \) |
| 31 | \( 1 + (6.53 - 7.49i)T + (-4.19 - 30.7i)T^{2} \) |
| 37 | \( 1 + (-1.20 - 1.61i)T + (-10.6 + 35.4i)T^{2} \) |
| 41 | \( 1 + (-0.459 + 1.78i)T + (-35.8 - 19.8i)T^{2} \) |
| 43 | \( 1 + (-7.48 - 6.79i)T + (4.16 + 42.7i)T^{2} \) |
| 47 | \( 1 + (-7.84 - 8.98i)T + (-6.36 + 46.5i)T^{2} \) |
| 53 | \( 1 + (-1.65 + 9.36i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-2.24 + 0.916i)T + (42.1 - 41.3i)T^{2} \) |
| 61 | \( 1 + (12.8 - 7.73i)T + (28.4 - 53.9i)T^{2} \) |
| 67 | \( 1 + (4.14 - 2.96i)T + (21.6 - 63.3i)T^{2} \) |
| 71 | \( 1 + (-1.59 - 5.32i)T + (-59.3 + 39.0i)T^{2} \) |
| 73 | \( 1 + (-8.65 + 2.05i)T + (65.2 - 32.7i)T^{2} \) |
| 79 | \( 1 + (-3.32 + 3.25i)T + (1.53 - 78.9i)T^{2} \) |
| 83 | \( 1 + (2.33 + 9.08i)T + (-72.6 + 40.0i)T^{2} \) |
| 89 | \( 1 + (-4.90 + 16.3i)T + (-74.3 - 48.9i)T^{2} \) |
| 97 | \( 1 + (9.63 + 0.374i)T + (96.7 + 7.51i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.56861560719803219490764562702, −9.536369841762595585086687511073, −8.879203722049512055722872049729, −7.51145493544754792606562922828, −6.41098670946179051403423577097, −5.88826706228066277372959826544, −5.23262074667289811195285505174, −4.17475104465701385220616251883, −2.96651396556955424227582158800, −1.82177277366703898019244369253,
1.93850500054828259775257784000, 2.54034867558937658111935805561, 3.93481047098660012093734474465, 4.99457764878144650965771630219, 5.61320356720926235889269350324, 6.42278651331975177589951303537, 7.37781466217035072600313520636, 9.025842852322310264889837333049, 9.474681546580762267896399423773, 10.69159912717008772031569053577
